Multiuser interference analysis of

MC-CDMA system using various spreading

sequences

Ki-Chun Cho

The Graduate School

Yonsei University

Department of Electrical and Electronic

Engineering

Multiuser interference analysis ofMC-CDMA system using various

spreading sequences

Ki-Chun Cho

A Thesis Submitted to the

Graduate School of Yonsei University

in Partial Fulfillment of the

Requirements for the Degree of

Master of Science

Supervised by

Professor Hong-Yeop Song, Ph.D.

Department of Electrical and Electronic EngineeringThe Graduate School

YONSEI University

December 2005

This certifies that the thesis ofKi-Chun Cho is approved.

Thesis Supervisor: Hong-Yeop Song

Jong-Moon Chung

Soo-Yong Choi

The Graduate SchoolYonsei UniversityDecember 2005

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Contents

List of Figures iv

List of Tables v

Abstract vi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 MC-CDMA system model 5

2.1 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 L-multipath channel model . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Multiuser interference and spectral correlation 9

3.1 The Detection model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Interference analysis according to combining methods .. . . . . . . . . 11

3.2.1 Maximal Ratio Combining (MRC) . . . . . . . . . . . . . . . . 11

i

3.2.2 Equal Gain Combining (EGC) . . . . . . . . . . . . . . . . . . 14

3.2.3 Orthogonality Restoring Combining (ORC) . . . . . . . . . .. 17

3.2.4 Minimum Mean Square Error Combining (MMSEC) . . . . . . 17

3.3 Spectral correlation according to interleaver . . . . . . .. . . . . . . . 19

3.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Channel model (1) . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.2 Channel model (2) . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Requirements of frequency spreading sequences 37

5 Spectral correlation profile 39

6 Concluding Remarks 46

Bibliography 49

Abstract (in Korean) 53

ii

List of Figures

2.1 MC-CDMA transmitter model . . . . . . . . . . . . . . . . . . . . . . 6

2.2 MC-CDMA receiver model . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Spectral correlation: |X(8,1)WH8

(l)| for Q = 1 . . . . . . . . . . . . . . . 20

3.2 Spectral correlation: |X(8,1)WH8

(l)| for Q = 4 . . . . . . . . . . . . . . . 20

3.3 BER performance using MRC, EGC (Users:4, Paths:2) . . . . .. . . . 23

3.4 BER performance using MMSEC, ORC (Users:4, Paths:2) . . .. . . . 23

3.5 Spectral correlation: X(6,5)WH8

(l) . . . . . . . . . . . . . . . . . . . . . . 24

3.6 Spectral correlation: X(7,5)WH8

(l) . . . . . . . . . . . . . . . . . . . . . . 24

3.7 Spectral correlation: X(8,5)WH8

(l) . . . . . . . . . . . . . . . . . . . . . . 24

3.8 Spectral correlation: X(3,2)WH8

(l) . . . . . . . . . . . . . . . . . . . . . . 25

3.9 Spectral correlation: X(5,2)WH8

(l) . . . . . . . . . . . . . . . . . . . . . . 25

3.10 Spectral correlation: X(8,2)WH8

(l) . . . . . . . . . . . . . . . . . . . . . . 25

3.11 BER performance using MRC, EGC (Users:4, Paths:4) . . . .. . . . . 26

3.12 BER performance using MMSEC, ORC (Users:4, Paths:4) . .. . . . . 26

3.13 BER performance for Walsh-Hadamard using MRC (Users:2) . . . . . 29

3.14 BER performance form-sequence Hadamard using MRC (Users:2) . . 29

iii

3.15 BER performance form-sequence Hadamard using MRC (Users:2) . . 30

3.16 BER performance for Walsh-Hadamard andm-sequence Hadamard us-

ing MMSEC (Users:4) . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.17 Spectral correlation:∑

u=2,3,4 X(u,1)WH8

(l) . . . . . . . . . . . . . . . . . 32

3.18 Spectral correlation:∑

u=3,5,8 X(u,2)WH8

(l) . . . . . . . . . . . . . . . . . 33

3.19 Spectral correlation:∑

u=3,7,8 X(u,1)MH8

(l) . . . . . . . . . . . . . . . . . 33

3.20 Spectral correlation:∑

u=2,3,5 X(u,1)MH8

(l) . . . . . . . . . . . . . . . . . 33

3.21 BER performance for Walsh-Hadamard andm-sequence Hadamard us-

ing MMSEC (Users:8) . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.22 Spectral correlation:∑

u 6=1 X(u,1)WH8

(l) . . . . . . . . . . . . . . . . . . 35

3.23 Spectral correlation:∑

u 6=1 X(u,1)MH8

(l) . . . . . . . . . . . . . . . . . . 35

5.1 BER performance of Zadoff-Chu and other sequence (Users:8, Paths:8) 43

5.2 Spectral correlation: X(3,1)Chu8

(l) . . . . . . . . . . . . . . . . . . . . . . 43

5.3 Spectral correlation: X(2,1)MH8

(l) . . . . . . . . . . . . . . . . . . . . . . 44

5.4 Spectral correlation: X(2,1)OG8

(l) . . . . . . . . . . . . . . . . . . . . . . 44

5.5 Spectral correlation: X(2,1)QH8

(l) . . . . . . . . . . . . . . . . . . . . . . 44

iv

List of Tables

3.1 Simulation parameters(1) . . . . . . . . . . . . . . . . . . . . . . . . .22

3.2 Simulation parameters(2) . . . . . . . . . . . . . . . . . . . . . . . . .27

3.3 ReqiredEb/No according to combining methods . . . . . . . . . . . . 27

3.4 Simulation parameters(3) . . . . . . . . . . . . . . . . . . . . . . . . .30

3.5 Maximum absolute value and requiredEb/No for target BER10−5 . . . 30

5.1 Simulation parameters(4) . . . . . . . . . . . . . . . . . . . . . . . . .43

v

ABSTRACT

Multiuser interference analysis of MC-CDMA systemusing various spreading sequences

Ki-Chun ChoDepartment of Electricaland Electronic Eng.The Graduate SchoolYonsei University

For effective communication over frequency selective fading channels, multicarrier

systems have been proposed as a scheme to enable high data rate transmission. Based on

this perspective, MC-CDMA systems, which is a combination of multicarrier systems

and CDMA, have been widely studied to achieve high user capacity. Applying frequency

diversity techniques based on spreading and combining datain the frequency domain can

results in a gain. When MC-CDMA technology is applied, frequency selectivity distorts

the amplitude and phase of the subcarriers, which breaks theorthogonality of the users

and as a result the multiuser interference increases. Therefore, effective control of the

interference could lead to a performance improvement.

MC-CDMA systems use frequency spreading sequences to separate multiusers. As

time spreading sequence characteristics affect the user capacity and system performance

of DS-CDMA, frequency spreading sequence characteristicsmay affect those of MC-

vi

CDMA systems. Spectral correlation is introduced as one of the methods to analyze

MUI with spreading sequences. This can be derived from a received signal expression,

and it differs from the conventional correlation properties which focus on time domain.

In multipath channels, the received signal can be expressedas a sum of many signals

multiplied by distinct channel coefficients. And this kind of distinct power and time de-

lay of multipath results in amplitude and phase variation ofspreading sequences, which

brings distortion of orthogonality among users. Spectral correlation shows features of

frequency spreading sequences including multipath time delay.

In this paper, we define the spectral correlation more clearly than conventional one to

be suited to downlink MC-CDMA system. And we can observe the expression of MUI

and control MUI with spreading sequence properties by usingthe spectral correlation.

L-multipath Rayleigh fading channels and multicarrier system are considered. MRC,

EGC, ORC, MMSEC are applied for combining methods. In this system, we observe

the relation between MUI and spectral correlation and derive the expression of MUI

based on sequence properties. And confirmation by simulation follows to investigate

system performance related to those properties. As a result, for MRC, it was shown that

MUI could be expressed as product of 3 terms: data symbol, channel elements, spectral

correlation, and it was observed that spectral correlationdirectly affected MUI. Here,

it was examined that the maximum absolute value of spectral correlation played a ma-

jor role to determine system performance, and good randomness of sequence made the

maximum absolute value small. Therefore, if spectral correlation atl′ was 0, correspond-

ing interference component became 0. And the system performances for 2 users using

Walsh-Hadamard andm-sequence Hadamard to reach target BER10−5 are examined.

vii

As a result, the best case using Walsh-Hadarard has 25.7dB and performance deviation

of 4.3dB, but the worst case usingm-sequence Hadamard has 25.7dB and performance

deviation of 0.9dB. For EGC, a similar conclusion could be obtained by approximation,

and performance differences relevant to spectral correlation were shown. For MMSEC,

the performance difference was also given. From the above spectral correlation proper-

ties, it can be known that the requirements of frequency spreading sequences are as fol-

lows: orthogonality of spreading sequences, small spectral correlation value, and good

randomness. And spectral correlation profile informs us of how the amplitude and phase

of the received signal are changed and how it affects the system performance. As a

result, the system using Zadoff-Chu sequence and BPSK modulation gained 3dB as op-

posed to using other sequences. Therefor, the MUI of MC-CDMAcan be estimated by

using spectral correlation, and sequences which is constructed based on the above fact

and allocation of selected sequences to some users can lead better system performance

and reduced deviation of system performances.

Key words : MC-CDMA, spectral correlation, multiuser interference, rayleigh fad-ing channel, frequency spreading sequences, combining,

viii

Chapter 1

Introduction

1.1 Motivation

CDMA (Code Division Multiple Access) is multiple access system which can transmit

many user’s signal simultaneously using code or sequence, and a multiple access sys-

tem based on direct sequence CDMA has achieved great importance for mobile radio

applications [1] [2]. Recently, CDMA technique has been considered to be a candi-

date to support multimedia services in mobile radio communications [3], because it has

its own capabilities to cope with asynchronous nature of multimedia data traffic, to pro-

vide higher capacity over conventional access techniques such as TDMA (Time-Division

Multiple Access) and FDMA (Frequency-Division Multiple Access).

On the other hand, an interesting approach to combat the distortions due to multipath

propagation in mobile communications based on multicarrier (MC) systems is consid-

ered. That system often called OFDM (Orthogonal frequency division multiplexing) is

applied to combat the frequency selectivity of the channel using a simple one tap equal-

izer. Furthermore OFDM prevents ISI (Inter Symbol Interference) and ICI (Inter Carrier

Interference) by inserting a guard interval between adjacent OFDM symbols [4] [5] [6].

1

The efficient combination of multicarrier modulation technique, i.e., OFDM with

CDMA, known as MC-CDMA has gained considerable attention both from academia

and industry as a promising technique for high datarate wireless communications sys-

tems [7] [8] [9]. MC-CDMA possesses the ability to mitigate severe multipath inter-

ference and the possibility to exploit the frequency diversity by spreading across sub-

carriers. Moreover, thanks to the guard interval between consecutive OFDM symbols,

MC-CDMA is an ISI free system as long as the delay spread of thechannel is shorter

than the guard interval. In addition to these properties, MC-CDMA possesses other ad-

vantages like efficient utilization of bandwidth, flexible resource management and ability

to generate different data rates within a fixed bandwidth [10]. However, through a fre-

quency selective fading channel, all the subcarriers have different amplitude levels and

different phase shifts (although they have high correlation among subcarriers), which

breaks the orthogonality among theses sequences, and resulting multiuser interference

(MUI) drastically reduces the system performance[8] [11] [12].

MC-CDMA system seriously suffers from performance degradation resulted from

MUI. Consequently, an effective scheme to manage MUI from analyzing it is required.

As time-domain autocorrelation and crosscorrelation function have been efficiently used

as criteria for measuring MUI for DS-CDMA systems, spreading sequences may affect

MC-CDMA system. Especially, selected sequence allocationin a sequence family may

cause performance difference [13]. But time-domain correlation functions are not proper

ways to look into MUI of MC-CDMA systems. Hence, it is essential to observe the

characteristic of spreading sequences including the effect of multipath delay.

There is a method to analyze the performance of MC-CDMA system, named as

2

spectral correlation function [14]. This can be evaluated from the expression of the

received signal in MC-CDMA system, which differs from the conventional time-domain

correlation function which has been applied for observing the correlation characteristic

among codes in DS-CDMA. In multipath channels, the receivedsignal is the sum of

transmitted signals passing through different paths. And that results in distortion of

codes. Spectral correlation function, a method of observing characteristics of codes

including multipath delay, reflects features of codes and time delay. Therefore, spectral

correlation could be a criterion measuring MUI in MC-CDMA system.

It is a goal of this paper to study a method suppressing MUI using spreading se-

quences from analyzing the relation between MUI and spreading sequences for MC-

CDMA. In this study, we consider L-multipath channel and MC-CDMA system, and

MRC, EGC, MMSEC, ORC which are well-known detection techniques for a single

user. In the above system, we analyze the relation between MUI and spreading sequence

using spectral correlation and how the characteristic of the sequence affects MUI. Also,

the requirements of spreading sequences to improve system performance are derived,

and what characteristic MUI has according to spreading sequence is observed.

1.2 Overview

Chapter 2 gives the description of MC-CDMA system model which covers a transmitter,

a receiver and L-multipath channel model. In Chapter 3, MUI analysis according to

4 combining methods and how to explain MUI using spectral correlation are carried

out. And spectral correlation is derived based on the general system model with serial-

to-parallel (S/P) and interleaver, and its physical meaning is investigated. And related

3

simulation results are shown. In Chapter 4, a summary of the requirements of codes to

mitigate MUI is given. In Chapter 5, we introduce a spectral correlation profile which

gives us knowledge of form of MUI. Finally, Chapter 6 draws conclusions and future

work.

4

Chapter 2

MC-CDMA system model

2.1 Transmitter

Figure 2.1 is MC-CDMA transmitter model. Input bit stream ismapped onto modulated

data symbols through modulation and pass a serial-to-parallel (S/P) block. TheQ num-

ber of data symbols are arranged in parallel and passQ number of copiers to be copiedN

number of chips respectively. The copiedN chips are multiplied by chips of frequency

spreading sequence of lengthN , c(u)n N−1

n=0 , where(u) represents a user. At this time,

total number of subcarriers isQN . Chip-interleaver takesN number of spreaded chips

of q-th data symbol on(q +Qn)-th subcarrier, wheren = 0, 1, 2, ..., N −1. Inverse Fast

Fourier Transform (IFFT) is applied to theseQN subcarriers and transmitted signal of

time durationT is generated. The complex baseband representation of the transmitted

signal on a(q + Qn)-th subcarrier,sq+Qn(t), in a signaling interval,T , can be written

as:

sq+Qn(t) =

√

Es

N

U−1∑

u=0

b(u)q c(u)

n ej2π(q+Qn)t/T , (2.1)

5

)(

0

uc

)(

1

u

Nc

)(

0

uc

)(

1

u

Nc

Q

Figure 2.1: MC-CDMA transmitter model

and total representation of the transmitted signal can be written as:

s(t) =

√

Es

N

U−1∑

u=0

Q−1∑

q=0

N−1∑

n=0

b(u)q c(u)

n ej2π(q+Qn)t/T , (2.2)

whereEs andb(u)q are the energy per data symbol and theq-th data symbol of(u)-th

user respectively.N is the length of frequency spreading sequence, andU is the number

of users.

2.2 L-multipath channel model

A wideband fading channel can be modelled as a sum of several differently delayed,

independent Rayleigh fading processes. The correspondingchannel impulse response

(CIR) is described as:

h(t, τ) =

L−1∑

l=0

al · hl(t)δ(τ − τl), (2.3)

whereal is the normalized amplitude such that∑L=1

l=0 a2l = 1.0, hl(t) is the Rayleigh

fading process withE[

|hl|2]

= 1.0, andδ(·) is the Dirac function.τlL−1l=0 is the delay

6

of the l-th path, andl is tap index of channel impulse response model [15].

For a multipath Rayleigh fading channel, the fading processcan be represented by

hl = αlejφl , whereαlL−1

l=0 , φlL−1l=0 are the random CIR tap amplitudes, phases,

respectively. We assume thatαlL−1l=0 , φlL−1

l=0 , τlL−1l=0 are mutually independent.

Fading amplitudesαlL−1l=0 are assumed to be statistically independent random variables

having a probability density function (PDF) expressed as:

f(αl) =2αl

Ωexp

(

−α2l

Ω

)

, (2.4)

whereΩ = E[α2l ]. The phasesφlL−1

l=0 of the different paths and of different subcarriers

are assumed to be uniformly distributed random variables in[0, 2π) [16], while the path

delay ofτlL−1l=0 are uniformly distributed in[0, Tmax], whereTmax is maximum delay

spread.

2.3 Receiver

A block diagram of the baseband model of the MC-CDMA receiverfor user(0) is rep-

resented on Figure 2.2. The signal received by user(0) during the symbol interval is

first OFDM-demodulated by applying an Fast Fourier Transform (FFT) of sizeQN and

despreaded with the(0)-th user’s spreading sequence.wnQN−1n=0 is a frequency do-

main equalization gain factor, which is dependent upon the employed diversity combin-

ing scheme. Combined data symbols are inserted into parallel-to-serial (P/S) block and

demapped to binary bit stream.

7

NQw

1

*)0(

0c

*)0(

1Nc

*)0(

0c

*)0(

1Nc

0w

1Nw

1QNw

Figure 2.2: MC-CDMA receiver model

8

Chapter 3

Multiuser interference and spectralcorrelation

In this chapter we investigate the received signal according to combining methods and

analyze the multiuser interference using spectral correlation. In section 3.1, we look into

the expression of the received signal. In section 3.2, multiuser interference according

to combining methods is derived and spectral correlation isdefined. And how spectral

correlation affects the multiuser interference and what meaning it has are derived. In

section 3.3, spectral correlation is applied to more general system with chip interleaver.

Simulation results supporting derived conclusion are showed in section 3.4.

3.1 The Detection model

From Figure 2.2, guard interval is removed from received signal and FFT is applied to

the signal. The FFT demodulated received symbolrq+QnN−1n=0

Q−1q=0 of the(q + Qn)-

th subcarrier can be expressed as:

rq+Qn =

√

Es

N

U−1∑

u=0

b(u)c(u)n Hq+Qn + Nq+Qn, (3.1)

9

whereHq+Qn is the(q + Qn)-th subcarrier’s frequency domain channel transfer factor,

andNq+Qn is a AWGN process having zero mean and a one-sided power spectral density

of No. In restoring arbitrary fixedq′-th data symbol amongQ number of parallel data

symbols, the decision variable of the(0)-th user’sq′-th data symbol,d(0)q′ , is given for a

single user detector as:

d(0)q′ =

N−1∑

n=0

wq′+Qn · c(0)∗n rq′+Qn, (3.2)

where(∗) represents the complex conjugation for the complex number.We assume

that there is no inter-subcarrier interference and frequency and timing synchronization

is accurate. The decision variabled(0)q′ can be expanded with the aid of Equation 3.1 and

3.2 as:

d(0)q′ =

N−1∑

n=0

wq′+Qn · c(0)∗n

(

√

Es

N

U−1∑

u=0

b(u)q′ c(u)

n Hq′+Qn + Nq′+Qn

)

= β + ζ + η, (3.3)

whereβ is the desired signal component given by

β =

√

Es

Nb(0)q′

N−1∑

n=0

Hq′+Qn · wq′+Qn, (3.4)

ζ is the MUI given by

ζ =

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n Hq′+Qn · wq′+Qn, (3.5)

andη is the noise component given by

η =

N−1∑

n=0

c(0)∗n Nq′+Qn · wq′+Qn. (3.6)

These three signal components predetermine the performance of the single user detector

considered[15].

10

3.2 Interference analysis according to combining methods

3.2.1 Maximal Ratio Combining (MRC)

Channel estimation technique makes it possible to obtain the frequency domain channel

transfer factor from which the frequency domain equalization gain factor can be derived.

The frequency domain channel transfer factor comes from FFTof the time domain chan-

nel impulse response and can be written as:

Hq+Qn =

QN−1∑

l=0

alhle−j2π(q+Qn)l/(QN)

=

QN−1∑

l=0

alhle−j2πql/(QN)e−j2πnl/N . (3.7)

WhenHq+QnN−1n=0

Q−1q=0 is frequency domain channel transfer factor, the equalization

gain factor,wq+Qn, for the MRC is given as:

wq+Qn = H∗q+Qn. (3.8)

We assume the perfect channel estimation, then the equalization gain factor, from Equa-

tion 3.7 and Equation 3.8, is given by:

wq+Qn =

QN−1∑

l=0

alhlej2πql/(QN)ej2πnl/N . (3.9)

11

Using above Equations,wq+QnHq+Qn can be rewritten as:

wq+QnHq+Qn = H∗q+QnHq+Qn

= a20h

20 + a2

1h21 + ... + a2

L−1h2L−1

+(

a0a1h0h∗1 + a1a2h1h

∗2 + ... + aL−2aL−1hL−2h

∗L−1

)

ej2πq1/QNej2πm1/N

+(

a0a1h∗0h1 + a1a2h

∗1h2 + ... + aL−2aL−1h

∗L−2hL−1

)

e−j2πq1/QNe−j2πm1/N

+(

a0a2h0h∗2 + a1a3h1h

∗3 + ... + aL−3aL−1hL−3h

∗L−1

)

ej2πq2/QNej2πm2/N

+(

a0a2h∗0h2 + a1a3h

∗1h3 + ... + aL−3aL−1h

∗L−3hL−1

)

e−j2πq2/QNe−j2πm2/N

...

+(

a0aL−1h0h∗L−1

)

ej2πq(L−1)/QNej2πm(L−1)/N

+ (a0aL−1h∗0hL−1) e−j2πq(L−1)/QNe−j2πm(L−1)/N

=L−1∑

l=−(L−1)

R(l) · ej2πnl/N , (3.10)

where

R(l) = ej2πql/(QN)L−1−l∑

k=0

akak+lhkh∗k+l, l = 0, 1, ..., L − 1, (3.11)

R(l) = ej2πql/(QN)L−1−l∑

k=0

akak−lh∗khk−l, l = −1, ...,−(L − 1). (3.12)

12

The corresponding user’s received signal component forq′-th data symbol,β, is given

by:

β =

√

Es

Nb(0)q′

N−1∑

n=0

|Hq′+Qn|2

=

√

Es

Nb(0)q′

N−1∑

n=0

L−1∑

l=−(L−1)

R(l)ej2πnl/N

=

√

Es

Nb(0)q′

L−1∑

l=−(L−1)

R(l)N−1∑

n=0

ej2πnl/N

=

√

Es

NR(0)b

(0)q′

N−1∑

n=0

1

=√

NEsb(0)q′ R(0). (3.13)

The MUI associated with MRC is given by:

ζ =

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

|Hq′+Qn|2c(u)n c(0)∗

n

=

√

Es

N

U−1∑

u=1

b(u)q′

L−1∑

l=−(L−)

R(l)

N−1∑

n=0

c(u)n c(0)∗

n ej2πnl/N . (3.14)

Now we define spectral correlation[14] between arbitrary different users(r) and(s)

which is given by:

X(r,s)(l) =N−1∑

n=0

c(r)n c(s)∗

n ej2πnl/N , (3.15)

and from the Equation 3.14 and 3.15,ζ can be expressed in the form

ζ =

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

|Hn|2c(u)n c(0)∗

n

=

√

Es

N

U−1∑

u=1

b(u)q′

L−1∑

l=−(L−1)

R(l)

N−1∑

n=0

c(u)n c(0)∗

n ej2πnl/N

=

√

Es

N

U−1∑

u=1

b(u)q′

L−1∑

l=−(L−1)

R(l)X(u,0)(l). (3.16)

13

As we can see in the Equation 3.16, MUI consists of 3 terms.

1) Different user’s data symbol,b(u)q′ ,

2) Channel element,R(l),

3) Spectral correlation,X(r,s)(l).

Because different user’s data symbol is not related to multipath fading, we exclude it

from considering MUI. Then MUI is regarded as product of the channel elements and

the spectral correlations. As we know from Equation 3.11 and3.12, channel element is

a sum of multipaths multiplied by other multipaths spacedTQN l time apart. Therefore,

spectral correlation,X(r,s)(l), can be regarded as a weight affecting to correlation value

between multipaths and other multipaths spacedTQN l time apart. In other word, spectral

correlation can influence characteristics of MUI and ifX(r,s)(l) is zero or small value,

then the corresponding interference can become zero or small value.

3.2.2 Equal Gain Combining (EGC)

The equalization gain factor,wq+Qn, for the EGC is given as:

wq+Qn =H∗

q+Qn

|Hq+Qn|. (3.17)

From Equation 3.17, the desired signal component of and MUI of q′-th data symbol are

given by:

β =

√

Es

Nb(0)q′

N−1∑

n=0

H∗q′+QnHq′+Qn

|Hq′+Qn|

=

√

Es

Nb(0)q′

N−1∑

n=0

√

H∗q′+QnHq′+Qn, (3.18)

14

ζ =

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n

H∗q′+QnHq′+Qn

|Hq′+Qn|

=

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n

√

H∗q′+QnHq′+Qn. (3.19)

We consider mean value and variance to analyze MUI. The mean value is given as:

E[H∗q+QnHq+Qn] = E

L−1∑

l=−(L−1)

R(l)ej2πnl/N

= E [R(0)] + E

L−1∑

l=−(L−1)l6=0

R(l)ej2πnl/N

= 1, (3.20)

and the variance is given as:

V ar[H∗q+QnHq+Qn] = V ar

L−1∑

l=−(L−1)

R(l)ej2πnl/N

= V ar [R(0)] + V ar

L−1∑

l=−(L−1)l6=0

R(l)ej2πnl/N

= V ar

[

L−1∑

k=0

a2kh

2k

]

+

L−1∑

l=−(L−1)l6=0

V ar

[

L−1−l∑

k=0

akak+lhkh∗k+l

]

= 2σ4L−1∑

k=0

a4k +

L−1∑

l=−(L−1)l6=0

σ4L−1−l∑

k=0

a2ka

2k+l

≤ 2a40σ

4L + 2a40σ

4 L(L − 1)

2

= a40σ

4L(L + 1). (3.21)

From the Taylor series, an expansion of√

x about 1 is given by:

√x ≈ 1 +

1

2(x − 1) − 1

8(x − 1)2 +

1

16(x − 1)3 − · · · . (3.22)

15

And the desired signal component and MUI approximated usingEquation 3.22 are given

as:

β ≈√

Es

Nb(0)q′

N−1∑

mn=0

(

1 +1

2(H∗

q′+QnHq′+Qn − 1)

)

=1

2

√

Es

Nb(0)q′ +

1

2

√

Es

Nb(0)q′

N−1∑

n=0

H∗q′+QnHq′+Qn

=1

2

√

Es

Nb(0)q +

1

2

√

NEsb(0)q R(0), (3.23)

ζ ≈√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n

(

1 +1

2(H∗

q′+QnHq′+Qn − 1)

)

=1

2

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n +1

2

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n H∗q′+QnHq′+Qn

=1

2

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n +1

2

√

Es

N

U−1∑

u=1

b(u)q′

L−1∑

l=−(L−1)

R(l)Xu,0(l). (3.24)

In the Equation 3.24, the first term is independent on the channel so that it can be consid-

ered as constant, especially zero when spreading sequencesare orthogonal. The second

term is dependent on the channel and similar to the expression of MRC, Equation 3.16.

Therefore, in the case of EGC, MUI can be expressed with spectral correlation like the

case of MRC and MUI characteristic is analogous to that of thecase of MRC, but smaller

than it by1/2. The energy of the desired signal is as large as that of MRC, however the

interference is smaller than that of MRC by1/2. Hence performance of EGC is better

than that of MRC.

16

3.2.3 Orthogonality Restoring Combining (ORC)

The equalization gain ,wq+Qn, for the ORC is given as:

wq+Qn =H∗

q+Qn

|Hq+Qn|2. (3.25)

From this Equation 3.25, the desired signal component and MUI of q′-th data symbol

are given by:

β =

√

Es

Nb(0)q′

N−1∑

n=0

H∗q′+QnHq′+Qn

|Hq′+Qn|2

=√

NEsb(0)q′ , (3.26)

ζ =

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n

H∗q′+QnHq′+Qn

|Hq′+Qn|2

=

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n . (3.27)

As shown in above equation, ORC cancels the effect of the channel so that spectral

correlation is not represented in the expression. Besides,if sequences are orthogonal,

MUI is equal to zero. But multipath diversity gain could not be acquired.

3.2.4 Minimum Mean Square Error Combining (MMSEC)

The equalization gain ,wq+Qn, for the MMSEC is given as:

wq+Qn =H∗

q+Qn

|Hq+Qn|2 + 2σ2

UNEs

. (3.28)

From this Equation 3.28, the desired signal component and MUI of q′-th data symbol

are given by:

β =

√

Es

Nb(0)q′

N−1∑

n=0

H∗q′+QnHq′+Qn

|Hq′+Qn|2 + 2σ2

UNEs

, (3.29)

17

ζ =

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n

H∗q′+QmHq′+Qm

|Hq′+Qn|2 + 2σ2

UNEs

. (3.30)

From the Taylor series, an expansion ofxx+a about 1 is given by:

x

x + a≈ 1

1 + a+

a

(1 + a)2(x−1)+

−2a

(1 + a)3(x−1)2+

6a

(1 + a)4(x−1)3+· · · , (3.31)

and the desired signal component and MUI approximated usingEquation 3.31 are given

as:

β ≈√

Es

Nb(0)q′

N−1∑

n=0

(

1

1 + 2σ2

UNEs

+2σ2

UNEs

(1 + 2σ2

UNEs

)2(H∗

q′+QnHq′+Qn − 1)

)

=

√

Es

Nb(0)q′

(

2σ2

UNEs

(1 + 2σ2

UNEs

)2

N−1∑

n=0

H∗q′+QnHq′+Qn +

N

(1 + 2σ2

UNEs

)2

)

=√

EsNb(0)q′

(

2σ2

UNEs

(1 + 2σ2

UNEs

)2R(0) +

1

(1 + 2σ2

UNEs

)2

)

, (3.32)

ζ ≈√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n

(

1

1 + 2σ2

UNEs

+2σ2

UNEs

(1 + 2σ2

UNEs

)2(H∗

q′+QnHq′+Qn − 1)

)

=

√

Es

N

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n

(

2σ2

UNEs

(1 + 2σ2

UNEs

)2H∗

q′+QnHq′+Qn +1

(1 + 2σ2

UNEs

)2

)

=

√

Es

N

(

1

(1 + 2σ2

UNEs

)2

)

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n

+

√

Es

N

(

2σ2

UNEs

(1 + 2σ2

UNEs

)2

)

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n H∗q′+QnHq′+Qn

=

√

Es

N

(

1

(1 + 2σ2

UNEs

)2

)

U−1∑

u=1

b(u)q′

N−1∑

n=0

c(u)n c(0)∗

n

+

√

Es

N

(

2σ2

UNEs

(1 + 2σ2

UNEs

)2

)

U−1∑

u=1

b(u)q′

L−1∑

l=−(L−1)

R(l)X(u,0)(l). (3.33)

In the Equation 3.33, the first term is independent on the channel so that it can be consid-

ered as constant, especially zero when spreading sequencesare orthogonal. The second

18

term is dependent on the channel and similar to the expression of MRC, Equation 3.16.

Therefore, in the case of MMSEC, MUI can be expressed with spectral correlation like

the case of MRC and MUI characteristic is analogous to that ofthe case of MRC. But

variance of the channel affects the MUI expression. Hence, the feature of MMSEC is

similar to that of ORC in high SNR, and to that of MRC in low SNR.Therefore, MUI is

slightly affected by spectral correlation.

3.3 Spectral correlation according to interleaver

In Equation 3.16, spectral correlation is a sum of products of reference user’s sequence,

c(0)∗n , and another user’s sequence,c

(u)n , andej2πnl/N , wherem = 0, 1, ..., N − 1. That

is the result which IFFT is applied to the product of two sequences andej2πnl/NN−1n=0 .

In that equation,Q, the size of S/P, is not shown. But back to the derivation, we can see

the spectral correlation includingQ which is given by:

X(r,s)(l′) =

N−1∑

n=0

c(r)n c(s)∗

n ej2πQnl′/QN . (3.34)

In other words, spectral correlation, Equation 3.15, is defined forQ = 1, and when

Q > 1, IFFT of sizeQN is applied to the product of sequences whose chip and another

chip are spacedQ-chip apart. Therefore, a spectral correlation forQ > 1 has a form of

repetition of the spectral correlation forQ = 1. Using above fact, the spectral correlation

for arbitraryQ > 1 andl′ ≥ N is given by:

X(r,s)(l′) = X(r,s)(l), l′ ≡ l(modN), l′ ≥ N > l. (3.35)

Figure 3.1 is a spectral correlation of Walsh-Hadamard of size 8 forQ = 1, and Figure

19

-6 -4 -2 0 2 4 6-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.1: Spectral correlation: |X(8,1)WH8

(l)| for Q = 1

-31 -24 -16 -8 0 8 16 24 31-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.2: Spectral correlation: |X(8,1)WH8

(l)| for Q = 4

20

3.2 is forQ = 4. For Q = 4, the spectral correlation forQ = 1 is repeated 4 times

sequentially. Therefore, regardless of S/P size,Q, we can estimate the feature of MUI

from only knowing the path delay interval index,l, and the spectral correlation,X(r,s)(l).

3.4 Simulation results

In this section, we investigate how BER performance of MC-CDMA system is affected

by spectral correlations. The channel is assumed to keep constant in one OFDM-CDMA

symbol and change from symbol to symbol independently. We assume perfect channel

estimation and accurate timing and frequency synchronization. And maximum delay

spread is within guard interval. We consider two channel model. First, CIR tap coef-

ficients are placed every sample point within guard intervaland exponentially decayed.

Second, CIR tap coefficients are placed arbitrary sample points within guard interval and

exponentially decayed.

3.4.1 Channel model (1)

Multipaths delay by one sample, and exponentially decay. 2 multipaths lie within[

0, TQN 1

]

or within guard interval. Figure 3.3 and Figure 3.4 show somesimulation results with

21

Walsh-Hadamard matrix of size 8 which is given by:

WH8 =

+ + + + + + + +

+ − + − + − + −+ + − − + + − −+ − − + + − − +

+ + + + − − − −+ − + − − + − +

+ + − − − − + +

+ − − + − + + −

(3.36)

and simulation parameters are shown on Table 3.1. In those figures, WH(5,6,7,8) means

that 5,6,7,8-th sequences of above Walsh-Hadamard matrix are allocated to different 4

users, respectively. Figure 3.3 is for MRC and EGC. BER performances of WH(5,6,7,8)

for MRC and EGC approach single user performances for MRC andEGC, which acquire

best performances. But BER performances of WH(2,3,5,8) forMRC and EGC are worse

than those of WH(5,6,7,8). Figure 3.4 is for MMSEC and ORC. BER performance

of WH(5,6,7,8) for MMSEC also approaches a single user performance for MMSEC,

which acquires best performance. But BER performance of WH(2,3,5,8) is worse than

that of WH(5,6,7,8). Because ORC cancels effect of the channel, there is no difference

according to spectral correlations.

Spectral correlations of Walsh-Hadamard of size 8 are shownon from Figure 3.5 to

Table 3.1: Simulation parameters(1)

FFT/IFFTPoints

Number ofsubcarriers

OFDM symbol Modulation QPSK

8 88+1 samples

(Guard interval=T/8)QPSK 8

22

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

o[dB]

BE

R

MRC single user

MRC (5.6.7.8) best

MRC (2.3.5.8) worse

EGC single user

EGC (5.6.7.8) best

EGC (2.3.5.8) worse

Figure 3.3: BER performance using MRC, EGC (Users:4, Paths:2)

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

o[dB]

BE

R

MMSEC single user

MMSEC (5.6.7.8) better

MMSEC (2.3.5.8) worse

ORC (5.6.7.8)

ORC (2.3.5.8)

Figure 3.4: BER performance using MMSEC, ORC (Users:4, Paths:2)

23

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-5

0

5

l

|X(l)|

Absolute Value

Figure 3.5: Spectral correlation: X(6,5)WH8

(l)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.6: Spectral correlation: X(7,5)WH8

(l)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.7: Spectral correlation: X(8,5)WH8

(l)

24

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.8: Spectral correlation: X(3,2)WH8

(l)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.9: Spectral correlation: X(5,2)WH8

(l)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.10: Spectral correlation: X(8,2)WH8

(l)

25

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

o[dB]

BE

R

MRC single user

MRC (5.6.7.8) better

MRC (2.3.5.8) worse

EGC single user

EGC (5.6.7.8) better

EGC (2.3.5.8) worse

Figure 3.11: BER performance using MRC, EGC (Users:4, Paths:4)

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

o[dB]

BE

R

MMSEC single user

MMSEC (5.6.7.8) better

MMSEC (2.3.5.8) worse

ORC single user

ORC better

ORC worse

Figure 3.12: BER performance using MMSEC, ORC (Users:4, Paths:4)

26

Table 3.2: Simulation parameters(2)

FFT/IFFTPoints

Number ofsubcarriers

OFDM symbol Modulation QPSK

32 3232+4 samples

(Guard interval=T/8)QPSK 8

Table 3.3: ReqiredEb/No according to combining methods

Number of paths:2 Number of path:4

10−5 singlebest

(no MUI)worst single best worst

MRC 24.9 24.9 36.1 16.5 41.0 x

EGC 25.3 25.7 32.5 17.5 23.8 x

MMSEC 25.0 25.7 27.1 16.7 21.0 21.8

Figure 3.10.X(r,s)WH8

(l) means spectral correlation of Walsh-Hadamard of size 8. From

Figure 3.5 to Figure 3.7,X(6,5)WH8

(±1) = X(7,5)WH8

(±1)= X(8,5)WH8

(±1) = 0. Those re-

sults mean that correlation between multipaths and other multipaths spacedTQN l time

apart becomes zero. In other words, MUI becomes zero so that the BER performances

approach single user performances. But from Figure 3.8 to Figure 3.10, because of

X(5,2)WH8

(±1) 6= 0 andX(8,2)WH8

(±1) 6= 0, MUI exist and BER performances of WH(2,3,5,8)

for MRC, EGC, MMSEC are degraded.

Figure 3.11 and Figure 3.12 show BER performances for 4 multipaths and sim-

ulation parameters are shown on Table 3.2. 4 multipaths delayed by one sample lie

within[

0, TQN 3

]

or within guard interval and are exponentially decayed. Forthese

cases, BER performances of WH(5,6,7,8) for MRC, EGC, MMSEC are better than

those of WH(2,3,5,8), too. BecauseX(6,5)WH8

(±l) = 0 for l = 0,±1,±2,±3 and oth-

ers have some values, not zero. Therefore, MUI for WH(5,6,7,8) get smaller than those

for WH(2,3,5,8), and the system using WH(5,6,7,8) acquiresbetter performance. Table

27

3.3 summarizes requiredEb/No[dB] for MC-CDMA systems that shown in this section

to attain BER10−5.

3.4.2 Channel model (2)

In this section, we consider more general case. 8 multipathsare within guard interval,

64 samples, and uniformly distributed in that and exponentially decayed.

Figure 3.13 and Figure 3.14 show BER performances of MC-CDMAsystem using

Walsh-Hadamard andm-sequence Hadamard for 2 users, and Figure 3.15 compares the

performances between those systems. And simulation parameters are shown on Table

3.4. m-sequence Hadamard matrix is obtained from the matrix consisting of all cyclic

shifts ofm-sequence by bordering the matrix on the top with a row of all zeros and on

the right by a column of all zeros. Them-sequence Hadamard matrix of size 8 we used

is given by:

MH8 =

+ + + + + + + +

+ + − + − − − +

+ − + − − − − +

− + − − − + + +

+ − − − + + − +

− − − + + − + +

− − + + − + − +

− + + − + − − +

(3.37)

The notation WH(1,2) means that we use 1-st and 2-nd row of Walsh-Hadamard

matrix, Equation 3.36, as spreading sequences of MC-CDMA system and allocate to

two different users respectively, and the notation MH(1,2)is for m-sequence Hadamard.

Figure 3.13, 3.14, and 3.15 show differences in BER performances according to selected

28

20 21 22 23 24 25 26 27 28 29 3010

-6

10-5

10-4

10-3

Eb/N

o[dB]

BE

R

WH(1,2).worst

WH(1,3).worse

WH(1,4).worse

WH(1,5).good

WH(1,6).good

Figure 3.13: BER performance for Walsh-Hadamard using MRC (Users:2)

20 21 22 23 24 25 26 27 28 29 3010

-6

10-5

10-4

10-3

Eb/N

o[dB]

BE

R

MH(1,8).worst

MH(1,2).good

MH(1,5).good

MH(1,7).good

Figure 3.14: BER performance form-sequence Hadamard using MRC (Users:2)

29

20 21 22 23 24 25 26 27 28 29 3010

-6

10-5

10-4

10-3

Eb/N

o[dB]

BE

R

WH(1,2).worst

WH(1,5).good

MH(1,8).worst

MH(1,2).good

Figure 3.15: BER performance form-sequence Hadamard using MRC (Users:2)

Table 3.4: Simulation parameters(3)

FFT/IFFTPoints

Number ofsubcarriers

OFDM symbol Modulation QPSK

512 512512+64 samples

(Guard interval=T/8)QPSK 8

Table 3.5: Maximum absolute value and requiredEb/No for target BER10−5

Pair ofsequences

WH(1,2)WH(1,3)WH(1,4)

WH(1,5)WH(1,8)

deviation MH(1,8)MH(1,2)MH(1,7)

deviation

Masimumabsolute value

8 5.6569 5.2263 2.7737 5.2263 4 1.2263

Largest/Smallestof MAV

largest smallest largest smallest

dB 30.0 27.2 25.7 4.3 25.7 24.8 0.9

30

sequences. Maximum of absolute value of spectral correlation can explain that. We de-

fine the value as maximum absolute value. Maximum absolute value is the largest value

among absolute values over−N < l < N in a spectral correlation, max−N<l<N

|X(r,s)(l)|

.

And in a sequence set, largest and smallest are exist among pairs of sequences. We call

the difference maximum absolute value deviation. Table 3.5shows maximum absolute

values and its maximum, and minimum values for Walsh-Hadamard andm-sequence

Hadamard. From Figure 3.13 and Table 3.5, we can observe thatgreater the maximum

absolute value, worse the BER performance. Same explanation can be applied tom-

sequence Hadamard.

Figure 3.15 compares both systems. From Figure 3.15, as WH(1,2) has largest max-

imum absolute value, it has worst BER performance, and as MH(1,2) has smallest max-

imum absolute value, it has best BER performance. And as WH(1,5) and MH(1,8) have

same maximum absolute value, we can observe that they have nearly same BER perfor-

mances. Judging from that, BER performance of MC-CDMA system for 2 users using

binary sequence depends on maximum absolute value of spectral correlation. Moreover,

for Walsh-Hadamard, largest maximum absolute value and smallest maximum absolute

value are 8 and 5.2263, respectively, and the difference between two values is 2.7737.

And for m-sequence Hadamard, largest and smallest value are 5.2263 and 4, respec-

tively, and the difference between two values is 1.2263 which is smaller than that for

Walsh-Hadamard. This influence BER performances shown in Figure 3.13, 3.14, and

3.15 and BER performance deviation. Therefore, because a sequence set having ran-

dom characteristic has small deviation of maximum absolutevalues, BER performance

deviation can be small. In Figure 3.16, BER performances of MC-CDMA system us-

31

20 20.5 21 21.5 22 22.5 23 23.5 2410

-6

10-5

10-4

Eb/N

o[dB]

BE

R

WH(1,2,3,4).worse

MH(2,3,5,8).worse

MH(1,3,7,8).better

WH(2,3,5,8).better

Figure 3.16: BER performance for Walsh-Hadamard andm-sequence Hadamard usingMMSEC (Users:4)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.17: Spectral correlation:∑

u=2,3,4 X(u,1)WH8

(l)

32

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.18: Spectral correlation:∑

u=3,5,8 X(u,2)WH8

(l)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.19: Spectral correlation:∑

u=3,7,8 X(u,1)MH8

(l)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.20: Spectral correlation:∑

u=2,3,5 X(u,1)MH8

(l)

33

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

o[dB]

BE

R

WH

MH

Figure 3.21: BER performance for Walsh-Hadamard andm-sequence Hadamard usingMMSEC (Users:8)

ing Walsh-Hadamard andm-sequence Hadamard for 4 users are shown and simulation

parameters are shown on Table 3.4. WH(1,2,3,4) means that 1,2,3,4-th row of Walsh-

Hadamard, Equation 3.36, are allocated to each user. From Figure 3.17 to Figure 3.20

show spectral correlations for 4 users. Spectral correlation for many users is expressed

as a sum of spectral correlation for 2 users and defined as:

∑

r 6=s

X(r,s)(l) =∑

r 6=s

N−1∑

n=0

c(r)n c(s)∗

n ej2πnl/N . (3.38)

From Figure 3.16, WH(1,2,3,4), MH(1,2,3,5), MH(1,3,7,8),WH(2,3,5,8) are in or-

der of bad performance. And WH(1,2,3,4) only has maximum absolute value 8, and

the others have 7.3910. Therefore WH(1,2,3,4) has the worstperformance, and the

34

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.22: Spectral correlation:∑

u 6=1 X(u,1)WH8

(l)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 3.23: Spectral correlation:∑

u 6=1 X(u,1)MH8

(l)

35

others are better than WH(1,2,3,4) and have little performance differences. Therefore

maximum absolute value of spectral correlation is major factor to decide BER perfor-

mance. Figure 3.21 shows BER performances using Walsh-Hadamard andm-sequence

Hadamard for 8 users, and simulation parameters are shown onTable 3.4. BER perfor-

mances of both systems are same for 8 users. This is due to samemaximum absolute

values of 8 for two sequences as shown in Figure 3.22 and Figure 3.23.

36

Chapter 4

Requirements of frequencyspreading sequences

From Equation 3.16, MUI is consists of channel elements and spectral correlation. First,

only to investigate effect of channel, we calculate mean andvariance of channel cross

product,R(l). Mean value is given by:

E [R(l)] =

1, for l = 0,

0, for l 6= 0.(4.1)

From the channel model, we assume that normalized amplitudes, alL−1l=0 , have ampli-

tudes ofa0 ≥ a1 ≥ · · · ≥ aL−1. Hence, variance is given by:

V ar [R(l)] = σ4L−1−l∑

k=0

a2ka

2k+l ≤ σ4a2

0a2l (L − 1). (4.2)

From Equation 4.1 and Equation 4.2, channel element forl = 0 has mean value 1

and larger variance than any other elements. Other elementshave mean value 0 and

smaller variance than 1 asl is larger. Therefore, spectral correlation atl = 0 should

be 0,X(r,s)(0) =∑N−1

m=0 c(r)m c

(s)∗m = 0, and this means that spreading sequences are

orthogonal. In Equation 3.24, first term of the expression can be 0, and in Equation 3.27,

37

interference can be 0, when spreading sequences are orthogonal. Hence, orthogonality

is most essential factor for frequency spreading sequences.

The first term of the MUI can be removed by using orthogonal sequences but the

other terms still remained. But those terms are consist of the channel elements and

spectral correlations so that each term can be controlled byspectral correlation value,

i.e., if a spectral correlation is zero or small, the term containing it is also zero or small.

If two multipaths are apaced by1T/QN , MUI can be 0 by using spreading sequence

whose spectral correlation at±1 is 0, Xr,s(±1) = 0. As we see simulation results at

secton 3.4, considering channel model and setting corresponding spectral correlation 0

or very small, MUI can be 0 or very small.

But if CIR tap coefficients are lied uniformly within guard interval larger thanN

samples, it is not enough to consider particular spectral correlation being 0 or small.

In this case, we consider not only spectral correlation at particular l, but also at all

−N < l < N . At that time, maximum absolute value of spectral correlation is regarded

as a major factor to system performance, and smaller it is, less MUI is, and better BER

performance. Random characteristic of spreading sequenceinfluence BER performance

difference according to user combinations. More random sequence is, smaller perfor-

mance deviation is.

38

Chapter 5

Spectral correlation profile

In this section, we investigate MUI more deeply and introduce spectral correlation pro-

file. Spectral correlation basically has complex value. Therefore, real part, imaginary

part, and absolute value of spectral correlation has various figures according to spreading

sequence. In this case,X(r,s)(l) andX(r,s)(−l) for l = 0, 1, ..., N − 1 can be complex

conjugate pair or not. We define spectral correlation profileas figure of spectral correla-

tion on complex plane.

From Equation 3.16 of MUI expression,R(l) and R(−l) are complex conjugate

pair. And MUI has form ofR(l)X(r,s)(l) +R(−l)X(r,s)(−l). Therefore, figure of MUI

can be changed by figures of spectral correlations,X(r,s)(l) andX(r,s)(−l). Now, we

consider two cases.

1) X(r,s)(−l) = X(r,s)∗(l),

2) EitherX(r,s)(l) or X(r,s)(−l) is zero.

For the case1), we useX(r,s)(−l) = X(r,s)∗(l) to express Equation 3.16, then MUI

39

is given by:

ζ1 =

√

ES

N

U−1∑

u=1

b(u)q′

L−1∑

l=1

2R

R(l)X(u,0)(l)

, (5.1)

whereR· is a function which results real value. In this case,R

R(l)X(u,0)(l)

has

only influence on real part of channel elements but imaginarypart. Therefore, magnitude

of MUI becomes twice of real part ofR(l)X(u,0)(l), and phase of MUI is distorted by

only data symbol.

For the case2), we useX(r,s)(−l) = 0 to express Equation 3.16, then MUI is given

by:

ζ1 =

√

ES

N

U−1∑

u=1

b(u)q′

L−1∑

l=1

R(l)X(u,0)(l). (5.2)

In this case,R(l)X(u,0)(l) has complex form, which has influence on both real and

imaginary part of channel elements. Therefore, magnitude of MUI becomes that of

R(l)X(u,0)(l), and phases of MUI is distorted by both data symbol andR(l)X(u,0)(l).

To investigate above phenomenon, some simulations using some sequences are per-

formed. We use Zadoff-Chu sequence, Walsh-Hadamard,m-sequence Hadamard, Or-

thogonal Gold sequence, Quadri-phase Hadamard, and introduce about those first.

Zadoff-Chu sequence [17] has optimum correlation properties. From that, orthog-

onal sequence set can be constructed. Zadoff-Chu orthogonal sequences of size 8 and

length 8 is given by:

ej2π

“

k2

2+qk

”

/2N, k = 0, 1, ..., N − 1, q = 0, 1, ..., N − 1, (5.3)

40

and for the simulation, we set the exponent as matrix(

k2

2 + qk)

, which is given by:

(

k2

2+ qk

)

=

0 3 12 11 0 11 12 3

0 9 8 13 8 9 0 13

0 15 4 15 0 7 4 7

0 5 0 1 8 5 8 1

0 11 12 3 0 3 12 11

0 1 8 5 8 1 0 5

0 7 4 7 0 15 4 15

0 13 0 9 8 13 8 9

. (5.4)

Orthogonal Gold sequence [18] is constructed usingm-sequence and its preferred

pair by using optimum autocorrelation property, and it is constructively similar tom-

sequence Hadamard. For the simulation, we use Orthogonal Gold sequence of size 8

given by:

OG8 =

+ + − + − − − +

+ + + − + − + +

+ − + − − + − +

− − + + + − − +

− − − − − − + +

− + + + − + + +

+ − − + + + + +

− + − − + + − +

. (5.5)

Quadri-phase Hadamard [19] of size 8 is 4-ary complex sequence having 4 symbols,

41

1,−1, j,−j. For the simulation, we use Quadri-phase Hadamard of size 8 given by:

QH8 =

+1 +j +1 +1 +j +j −j −1

+1 −j +1 +1 −j −j +j −1

+1 +j −1 +1 +j −j +j +1

+1 −j −1 +1 −j +j −j +1

+1 +j +1 −1 −j −j −j +1

+1 −j +1 −1 +j +j +j +1

+1 +j −1 −1 −j +j +j −1

+1 −j −1 −1 +j −j −j −1

. (5.6)

Figure 5.1 shows BER performances whose modulation methodsare BPSK, and Table5.1

shows simulation parameters of the systems. We can observe that the system using

Zadoff-Chu sequence gains 3dB at target BER10−5. This comes from the fact that

spectral correlation profiles of binary sequence and Quadri-phase Hadamard belong to

the case1) previous mentioned, but that of Zadoff-Chu sequence belongs to the case

2). While using BPSK modulation which has data symbol as+1,−1, we decide

the sign of data symbol only on real part. For the case1), MUI is under influence

of 2R

R(l)X(u,0)(l)

, while for the case2),R

R(l)X(u,0)(l)

. In other words, for

the system which use Zadoff-Chu sequence and decide the signof data symbol on 1-

dimension like BPSK, MUI reduced by half.

Figure 5.2, 5.3, 5.4, and 5.5 are spectral correlation of Zadoff-Chu sequence,m-

sequence Hadamard, Orthogonal Gold, and Quadri-phase Hadamard. As we can see

the figures,X(r,s)Chu8

(−2) = 8, X(r,s)Chu8

(2) = 0, thus Zadoff-Chu belongs to the case2).

Others areX(r,s)(−l) =(

X(r,s)(l))∗

and belong to the case1). In fact, all binary

42

Table 5.1: Simulation parameters(4)

FFT/IFFTPoints

Number ofsubcarriers

OFDM symbol Modulation QPSK

64 6464+8 samples

(Guard interval=T/8)BPSK 8

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

o[dB]

BE

R

Zadoff-Chu

Walsh-Hadamard

m-sequence Hadamard

Orthogonal Gold

Quadi-phase Hadamard

Figure 5.1: BER performance of Zadoff-Chu and other sequence (Users:8, Paths:8)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 5.2: Spectral correlation: X(3,1)Chu8

(l)

43

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 5.3: Spectral correlation: X(2,1)MH8

(l)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 5.4: Spectral correlation: X(2,1)OG8

(l)

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ReX

(l)

Real

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

ImX

(l)

Imaginary

-5 0 5-8

-6

-4

-2

0

2

4

6

8

l

|X(l)|

Absolute Value

Figure 5.5: Spectral correlation: X(2,1)QH8

(l)

44

integer sequences are in case1) because of below expression as:

X(r,s)(−l) =

N−1∑

m=0

c(r)m c(s)∗

m expj2πm(−l)/N

=

(

N−1∑

m=0

c(r)∗m c(s)

m expj2πml/N

)∗

= X(r,s)∗(l). (5.7)

Quadri-phase Hadamard sequence is a nonbinary, complex sequence. But it hasc(r)m c

(s)∗m =

c(r)∗m c

(s)m so that it belongs to the case1). In other words, we can see how MUI appears

as observing spectral correlation profile.

45

Chapter 6

Concluding Remarks

In this paper, we have analyzed the relation between a multiuser interference and fre-

quency spreading sequences in a MC-CDMA system by using a spectral correlation.

Based on this, we investigated requirements of frequency spreading sequence and what

figure MUI takes on according to the spectral correlation profile. MRC, EGC, ORC,

and MMSEC were considered as combining methods of the MC-CDMA system for sin-

gle user detection as well as an L-multipath Rayleigh fadingchannel. As a result of

examining MUI using 4 methods, MUI of the system using MRC wasdrawn as prod-

uct of 3 terms: data symbol, channel elements, and spectral correlation. And we veri-

fied that MUI of the system using EGC and MMSEC had a similar form to that using

MRC through approximation and simulation. In the system with S/P and interleaver, we

showed that MUI is also derived from spectral correlation.

Spectral correlation is weight which has an effect on interference so thatX(r,s)(±l)

directly affects products of multipaths spacedτl apart. Thus, we verified that if spectral

correlation at somel is 0, the corresponding interference could be 0 by derivation and

simulation. In order to confirm this, we have observed the requiredEb/No[dB] for target

46

BER 10−5 in the channel which has 2 multipaths spacedTQN apart by using 4 different

Walsh-Hadamard sequences, which haveX(r,s)(±1) = 0. As a result, the system for a

single user using MRC, EGC, and MMSEC needed 24.9, 25.3, and 25 dB, respectively,

and for 4 users, about 24.9, 25.7, and 25.7 dB, respectively.These show that there are

nearly no interferences.

The maximum absolute value of spectral correlation is a major factor affecting an

interference. Smaller the maximum absolute value is, the better the system performance,

and smaller the deviation of maximum absolute values is, thesmaller the deviation of

system performances according to user allocation is. This relates to the randomness of

sequences so that sequences which have good randomness havesmall maximum absolute

values and small deviation of maximum absolute values. A typical binary sequence

which has good randomness ism-sequence Hadamard.m-sequence Hadamard of size

8 and length 8 has smallest the maximum absolute value of 4, which is smaller than

that of Walsh-Hadamard of 5.2263, and it has the largest maximum absolute value of

5.2263, which is smaller than that of Walsh-Hadamard of 8, and it has a smaller deviation

of maximum absolute values. Thus, when the sequences are allocated to some users,

the system usingm-sequence Hadamard is better than that of Walsh-Hadamard, and

has a smaller deviation of performances. In order to confirm this, we have observed

the requiredEb/No[dB] for target BER10−5 in the channel which has 8 multipaths

spaced arbitrary apart within the guard interval. A system using MRC andm-sequence

Hadamard for 2 users needed 24.8 and 25.7dB with respect to the maximum absolute

values of 4 and 5.2263, respectively, and the one using Walsh-Hadamard needed 25.7

and 30.3dB with respect to the maximum absolute values of 5.2263 and 8, respectively.

47

A system using MMSEC andm-sequence Hadamard for 4 users needed about 21.4dB

with respect to both the largest and the smallest maximum absolute values and the one

using Walsh-Hadamard needed 21.6 and 21.7 dB with respect tothe largest and the

smallest maximum absolute values.

As a consequence of the above results, we mention the requirements of frequency

spreading sequences. First, the orthogonality of sequences to distinguish multiusers

is essential. From the derived MUI expression, the largest interference factor can be

removed by using the orthogonal sequence. Second, spectralcorrelation should be 0 or

small. From the expression, spectral correlation directlyaffects channel elements, which

also affects the interference. Third, good randomness of a sequence is required. The

more random a sequence is, the smaller the spectral correlation value and the deviation

of system performance and the better the system performanceare in case of allocation

of selected sequences to some users is carried out.

On the other hand, spectral correlation profile offers us what form MUI takes and

how it affects the system. All binary sequences and Quadri-phase Hadamard used in

this paper have same spectral correlation profile, but Zadoff-Chu sequence has a differ-

ent spectral correlation profile. Therefore, in a system which performs 1-dimensional

decision like BPSK, the interference affecting the decision is reduced. In order to con-

firm this, we have observed the requiredEb/No[dB] for target BER10−6 using BPSK

modulation. In that simulation, the system using Zadoff-Chu sequence gained 3dB as

opposed to using other sequences.

MC-CDMA which would use advantages of OFDM and CDMA is widelystudied

for high data rate multimedia communication. But MUI greatly degrades system perfor-

48

mance. In this paper, spectral correlation and spectral correlation profile are introduced

as a method to manage and analyze MUI. And we verified it by derivation and simula-

tion. The MUI of MC-CDMA can be estimated by using spectral correlation. Therefore,

sequences which is constructed based on the above fact and allocation of selected se-

quences to some users can lead better system performance andreduced deviation of

system performances.

Binary sequences are mainly dealt in this study; complex sequences, however, have

more various appearances which affect spectral correlation and system performance than

binary sequences do. Therefore, examining the relation between spectral correlation of

complex sequences and system performance would contributeto development of system

performance.

49

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52

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